# Locally testable codes and PCPs of almost-linear length

@article{Goldreich2006LocallyTC,
title={Locally testable codes and PCPs of almost-linear length},
journal={Electron. Colloquium Comput. Complex.},
year={2006}
}
• Published 1 July 2006
• Computer Science, Mathematics
• Electron. Colloquium Comput. Complex.
We initiate a systematic study of locally testable codes; that is, error-correcting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject non-codewords with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we…
164 Citations

## Topics from this paper

Locally testable codes via high-dimensional expanders
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2020
This work unifies and generalizes the known results on testability of the Hadamard, Reed-Muller and lifted codes on the Subspace Complex, all of which are proved via local self correction by performing iterative self correction with logarithmically many rounds and tightly controlling the error in each iteration using properties of the high-dimensional expander.
Almost orthogonal linear codes are locally testable
• Mathematics, Computer Science
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
• 2005
The condition to show that almost-orthogonal codes are locally testable, and their dual codes can be spanned by words of constant weights is used, which can be straightforwardly extended to Goppa codes and trace subcodes of algebraic-geometric codes.
Sparse Random Linear Codes are Locally Decodable and Testable
• Computer Science, Mathematics
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
• 2007
The results are the first to show that local decodability and testability can be found in random, unstructured, codes and derive the local testability of linear codes from the classical coding theory parameters, namely the rale and the distance of the codes.
Error correcting codes: local testing, list decoding, and applications
• Mathematics
• 2007
This dissertation is a study of special types of error correcting codes and their applications. It consists of three parts. First, we study Generalized Reed-Muller codes (over prime fields), aka
High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2015
This work constructs LTCs that have high rate (arbitrarily close to 1), have constant relative distance, and can be tested using (log n) O(log logn) queries, which improves over the previous best construction of L TCs with high rate, by the same authors.
Locally Testable Codes Require Redundant Testers
• Computer Science, Mathematics
2009 24th Annual IEEE Conference on Computational Complexity
• 2009
This paper proves the stronger claim that the {\em actual test itself} must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant local testing is impossible.
Locally Testable Codes Require Redundant Testers
• Mathematics, Computer Science
Computational Complexity Conference
• 2009
This paper proves the stronger claim that the {\em actual test itself} must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant local testing is impossible.
Strong Locally Testable Codes with Relaxed Local Decoders
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2014
This work constructs a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs, which improves upon the previously known constructions, which either obtain weak LTCs or require polynomial length.
Strong Locally Testable Codes with Relaxed Local Decoders
• Computer Science, Mathematics
ACM Trans. Comput. Theory
• 2019
This work constructs a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs, which improves upon the previously known constructions, which either obtain only weak LTCs or require polynomial length.
Low Rate Is Insufficient for Local Testability
• Computer Science, Mathematics
APPROX-RANDOM
• 2010
The last result shows that redundancy is indeed a function of the code dimension, not blocklength, and that the bound given in [1] is nearly tight.

## References

SHOWING 1-10 OF 80 REFERENCES
Locally testable codes and PCPs of almost-linear length
• Mathematics, Computer Science
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
• 2002
It is shown that certain PCP systems can be modified to yield locally testable codes and PCPs of almost-linear length, and novel techniques in use include a random projection of certain codewords and PCP-oracles, an adaptation of PCP constructions to obtain "linear PCP -oracles" for proving conjunctions of linear conditions, and a direct construction of local testable (linear) codes of sub-exponential length.
Bounds on 2-Query Codeword Testing
• Computer Science, Mathematics
RANDOM-APPROX
• 2003
We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2-locally testable code with minimal distance δn over any
Simple PCPs with poly-log rate and query complexity
• Computer Science, Mathematics
STOC '05
• 2005
This work shows how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed-Solomon codeword, i.e., a univariate polynomial of specified degree.
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
• Mathematics, Computer Science
STOC '03
• 2003
We present the first explicit construction of Probabilistically Checkable Proofs (PCPs) and Locally Testable Codes (LTCs) of fixed constant query complexity which have almost-linear (= n * 2Õ(√log
Lower bounds for linear locally decodable codes and private information retrieval
• Computer Science
Proceedings 17th IEEE Annual Conference on Computational Complexity
• 2002
We prove that if a linear error-correcting code C: {0, 1}/sup n/ /spl rarr/ {0, 1}/sup m/ is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a
Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding
• Mathematics, Computer Science
SIAM J. Comput.
• 2006
The main technical contribution is a construction of a “length-efficient” robust PCP of proximity, which does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al.
A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP
• Mathematics, Computer Science
STOC '97
• 1997
A new low-degree-test is introduced, one that uses the restriction of low- degree polynomials to planes rather than the restriction to lines, and enables us to prove a low-error characterization of NP in terms of PCP.
Improved low-degree testing and its applications
• Computer Science, Mathematics
STOC '97
• 1997
A new, and surprisingly strong, analysis is presented which shows that the preceding statement is true for arbitrarily small δ, provided the field size is polynomially larger than d/δ, and produces a self tester/corrector for any buggy program that computes a polynomial over a finite field.
Testing and Weight Distributions of Dual Codes
• Marcos A. Kiwi
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 1997
This framework establishes a connection between testing and the theory of weight distributions of dual codes, and derives from the MacWilliams Theorems a general result, the Duality Testing Lemma, and uses it to analyze the simpler tests that fall into this framework.
Free Bits, PCPs, and Nonapproximability-Towards Tight Results
• Mathematics, Computer Science
SIAM J. Comput.
• 1998
A "reversal" of the connection between probabilistically checkable proofs (PCPs) and the approximability of NP-optimization problems is presented and any NP-hardness of approximation result for MaxClique yields a proof system for NP.